Topological Cluster Statistics (TCS)¶

Notebook 6: informedness evaluations¶

This notebook contains scripts that evaluate the informedness of TCS compared to cluster-based statistic. Informedness (bookmakers informedness, also known as Youden's J statistic), is a measure equivalent to the balanced accuracy of a classifier and evaluates if the increase in sensitivity was at the cost of introducing inflated false positives.

Packages and basic functions¶

Loading required packages

In [1]:
import os
import numpy as np
import pandas as pd
import nibabel as nib
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn import metrics
from tqdm.notebook import tqdm

Basic functions

In [2]:
def ensure_dir(file_name):
    os.makedirs(os.path.dirname(file_name), exist_ok=True)
    return file_name


def write_np(np_obj, file_path):
    with open(file_path, 'wb') as outfile:
        np.save(outfile, np_obj)


def load_np(file_path):
    with open(file_path, 'rb') as infile:
        return np.load(infile)

Plot settings (latex is used for better plotting)

In [3]:
sns.set()
sns.set_style("darkgrid")

%matplotlib inline
%config InlineBackend.figure_format = 'retina'
In [4]:
plt.rc('text', usetex=True)
plt.rc('text.latex', preamble=r'\usepackage{mathtools} \usepackage{sfmath}')

plt.rc('xtick', labelsize=20)
plt.rc('ytick', labelsize=20)
plt.rc('axes', labelsize=24)

plt.rc('figure', dpi=500)

Loading the ground truth¶

The ground truth stored in notebook 2 is loaded here.

In [5]:
# list of all tasks and the cope number related to each selected contrast
tasks = {
    'EMOTION': '3',  # faces - shapes
    'GAMBLING': '6',  # reward - punish
    'RELATIONAL': '4',  # rel - match
    'SOCIAL': '6',  # tom - random
    'WM': '20',  # face - avg
}
In [6]:
# Compute mean and std, followed by a parametric z-score (one sample t-test)
ground_truth_effect = {}
# Base directory where files are stored at
base_dir='/data/netapp01/work/sina/structural_clustering/PALM_revision_1'

for task in tqdm(tasks, desc="Tasks loop", leave=True):
    ground_truth_effect[task] = load_np(
        '{}/ground_truth/cohen_d_{}_cope{}.dscalar.npy'.format(base_dir, task, tasks[task]),
    )

Tasks loop:   0%|          | 0/5 [00:00<?, ?it/s]

Loading PALM results¶

PALM results stored in notebook 1 is loaded here.

In [7]:
%%time

# Number of random repetitions
repetitions = 500
# Different sample sizes tested
sample_sizes = [10, 20, 40, 80, 160, 320]
# Different cluster defining thresholds
cdts = [3.3, 2.8, 2.6, 2.0, 1.6]
# Number of brainordinates in a cifti file
Nv = 91282
# Base directory where files are stored at
base_dir='/data/netapp01/work/sina/structural_clustering/PALM_revision_1'

# Store loaded results in nested python dictionaries
loaded_maps = {}
loaded_maps['uncorrected_tstat'] = {}
loaded_maps['spatial_cluster_corrected_tstat'] = {}
loaded_maps['topological_cluster_corrected_tstat'] = {}

# Only use the z=3.3, p=0.001 for the main analyses reported here
cdt = 3.3
sample_size = 40
for task in tqdm(tasks, desc="Tasks loop", leave=True):
    loaded_maps['uncorrected_tstat'][task] = {}
    loaded_maps['spatial_cluster_corrected_tstat'][task] = {}
    loaded_maps['topological_cluster_corrected_tstat'][task] = {}
    loaded_maps['uncorrected_tstat'][task][f'N={sample_size}'] = load_np(
        f'{base_dir}/summary/uncorrected_tstat_{task}_{sample_size}_samples_{cdt}_CDT.npy',
    )
    loaded_maps['spatial_cluster_corrected_tstat'][task][f'N={sample_size}'] = load_np(
        ensure_dir(f'{base_dir}/summary/spatial_cluster_corrected_tstat_{task}_{sample_size}_samples_{cdt}_CDT.npy'),
    )
    loaded_maps['topological_cluster_corrected_tstat'][task][f'N={sample_size}'] = load_np(
        ensure_dir(f'{base_dir}/summary/topological_cluster_corrected_tstat_{task}_{sample_size}_samples_{cdt}_CDT.npy'),
    )

Tasks loop:   0%|          | 0/5 [00:00<?, ?it/s]
CPU times: user 45.3 ms, sys: 7.08 s, total: 7.13 s
Wall time: 31.2 s

Informedness analyses¶

Script below generates the results of informedness analysis reported in the manuscript.

For more information, refer to the following:

  • https://biodatamining.biomedcentral.com/articles/10.1186/s13040-021-00244-z#:~:text=Bookmaker%20informedness%20is%20the%20only,would%20have%20MCC%20%3D%20%2B1.
  • https://www.sciencedirect.com/science/article/pii/S016786552030115X
In [8]:
%%time

# number of repititions for every sample size (repeated subsampling)
repetitions = 500
Nv = 91282

logp_threshold = -np.log10(0.05)
ground_truth_effect_thresholds = np.linspace(0.01,0.2,)

# bookmarker informedness
BM = {}
TPR = {}
FPR = {}

for task in tqdm(tasks, desc="Tasks loop", leave=True):
    BM[task] = {}
    TPR[task] = {}
    FPR[task] = {}
    sample_size = 40

    BM[task][f'N={sample_size}'] = {}
    TPR[task][f'N={sample_size}'] = {}
    FPR[task][f'N={sample_size}'] = {}
    BM[task][f'N={sample_size}']['spatial'] = []
    BM[task][f'N={sample_size}']['topological'] = []
    TPR[task][f'N={sample_size}']['spatial'] = []
    TPR[task][f'N={sample_size}']['topological'] = []
    FPR[task][f'N={sample_size}']['spatial'] = []
    FPR[task][f'N={sample_size}']['topological'] = []

    t_stats = loaded_maps['uncorrected_tstat'][task][f'N={sample_size}']
    t_stats = t_stats[~np.isnan(t_stats).any(axis=1)]

    spatial_cluster_logps = loaded_maps['spatial_cluster_corrected_tstat'][task][f'N={sample_size}']
    spatial_cluster_logps = spatial_cluster_logps[~np.isnan(spatial_cluster_logps).any(axis=1)]
    # statistical predictions
    spatial_increased_activation = (spatial_cluster_logps>logp_threshold) & (t_stats>0)
    spatial_decreased_activation = (spatial_cluster_logps>logp_threshold) & (t_stats<0)
    spatial_no_change_in_activation = (spatial_cluster_logps<logp_threshold)

    topological_cluster_logps = loaded_maps['topological_cluster_corrected_tstat'][task][f'N={sample_size}']
    topological_cluster_logps = topological_cluster_logps[~np.isnan(topological_cluster_logps).any(axis=1)]
    # statistical predictions
    topological_increased_activation = (topological_cluster_logps>logp_threshold) & (t_stats>0)
    topological_decreased_activation = (topological_cluster_logps>logp_threshold) & (t_stats<0)
    topological_no_change_in_activation = (topological_cluster_logps<logp_threshold)

    for ground_truth_effect_threshold in tqdm(ground_truth_effect_thresholds, desc="Thresholds loop", leave=False):
        # ground truth conditions
        true_increased_activation = ground_truth_effect[task] > ground_truth_effect_threshold
        true_decreased_activation = ground_truth_effect[task] < -ground_truth_effect_threshold
        true_no_change_in_activation = np.abs(ground_truth_effect[task]) < ground_truth_effect_threshold

        spatial_true_positive_increased_activation = np.multiply(spatial_increased_activation, np.tile(true_increased_activation,(spatial_cluster_logps.shape[0],1))).sum(1)
        spatial_true_positive_decreased_activation = np.multiply(spatial_decreased_activation, np.tile(true_decreased_activation,(spatial_cluster_logps.shape[0],1))).sum(1)
        spatial_true_positive = spatial_true_positive_increased_activation + spatial_true_positive_decreased_activation
        spatial_true_negative = np.multiply(spatial_no_change_in_activation, np.tile(true_no_change_in_activation,(spatial_cluster_logps.shape[0],1))).sum(1)
        spatial_false_negative_increased_activation = np.multiply(spatial_no_change_in_activation, np.tile(true_increased_activation,(spatial_cluster_logps.shape[0],1))).sum(1)
        spatial_false_negative_decreased_activation = np.multiply(spatial_no_change_in_activation, np.tile(true_decreased_activation,(spatial_cluster_logps.shape[0],1))).sum(1)
        spatial_false_negative = spatial_false_negative_increased_activation + spatial_false_negative_decreased_activation
        spatial_false_positive = Nv - (spatial_true_positive + spatial_true_negative + spatial_false_negative)
        spatial_true_positive_rate = np.divide(spatial_true_positive, (spatial_true_positive + spatial_false_negative))
        spatial_false_positive_rate = np.divide(spatial_false_positive, (spatial_false_positive + spatial_true_negative))
        spatial_BM = spatial_true_positive_rate - spatial_false_positive_rate
        TPR[task][f'N={sample_size}']['spatial'].append(spatial_true_positive_rate)
        FPR[task][f'N={sample_size}']['spatial'].append(spatial_false_positive_rate)
        BM[task][f'N={sample_size}']['spatial'].append(spatial_BM)

        topological_true_positive_increased_activation = np.multiply(topological_increased_activation, np.tile(true_increased_activation,(topological_cluster_logps.shape[0],1))).sum(1)
        topological_true_positive_decreased_activation = np.multiply(topological_decreased_activation, np.tile(true_decreased_activation,(topological_cluster_logps.shape[0],1))).sum(1)
        topological_true_positive = topological_true_positive_increased_activation + topological_true_positive_decreased_activation
        topological_true_negative = np.multiply(topological_no_change_in_activation, np.tile(true_no_change_in_activation,(topological_cluster_logps.shape[0],1))).sum(1)
        topological_false_negative_increased_activation = np.multiply(topological_no_change_in_activation, np.tile(true_increased_activation,(topological_cluster_logps.shape[0],1))).sum(1)
        topological_false_negative_decreased_activation = np.multiply(topological_no_change_in_activation, np.tile(true_decreased_activation,(topological_cluster_logps.shape[0],1))).sum(1)
        topological_false_negative = topological_false_negative_increased_activation + topological_false_negative_decreased_activation
        topological_false_positive = Nv - (topological_true_positive + topological_true_negative + topological_false_negative)
        topological_true_positive_rate = np.divide(topological_true_positive, (topological_true_positive + topological_false_negative))
        topological_false_positive_rate = np.divide(topological_false_positive, (topological_false_positive + topological_true_negative))
        topological_BM = topological_true_positive_rate - topological_false_positive_rate
        TPR[task][f'N={sample_size}']['topological'].append(topological_true_positive_rate)
        FPR[task][f'N={sample_size}']['topological'].append(topological_false_positive_rate)
        BM[task][f'N={sample_size}']['topological'].append(topological_BM)

    TPR[task][f'N={sample_size}']['spatial'] = np.array(TPR[task][f'N={sample_size}']['spatial'])
    FPR[task][f'N={sample_size}']['spatial'] = np.array(FPR[task][f'N={sample_size}']['spatial'])
    BM[task][f'N={sample_size}']['spatial'] = np.array(BM[task][f'N={sample_size}']['spatial'])

    TPR[task][f'N={sample_size}']['topological'] = np.array(TPR[task][f'N={sample_size}']['topological'])
    FPR[task][f'N={sample_size}']['topological'] = np.array(FPR[task][f'N={sample_size}']['topological'])
    BM[task][f'N={sample_size}']['topological'] = np.array(BM[task][f'N={sample_size}']['topological'])
        

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CPU times: user 1min 20s, sys: 2.29 s, total: 1min 22s
Wall time: 1min 22s
In [9]:
def get_cohen_d_from_bonferroni_corrected_p_value(p, sample_size, multiple_comparisons):
    # only for two tailed T test
    unc_p = p / multiple_comparisons
    t_stat = stats.t.isf(unc_p/2, sample_size-1)
    cohen_d = t_stat / np.sqrt(sample_size)
    return cohen_d

def get_bonferroni_corrected_p_value_from_cohen_d(d, sample_size, multiple_comparisons):
    t_stat = d * np.sqrt(sample_size)
    unc_p = stats.t.sf(np.abs(t_stat), sample_size-1)*2
    p = unc_p * multiple_comparisons
    return p

def get_uncorrected_p_value_from_cohen_d(d, sample_size, multiple_comparisons):
    t_stat = d * np.sqrt(sample_size)
    unc_p = stats.t.sf(np.abs(t_stat), sample_size-1)*2
    return unc_p

def compute_normalized_partial_area_under_curve(fpr, tpr, lowest_fpr, highest_fpr):
    fpr_limited = np.sort(fpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    tpr_limited = np.sort(tpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    return (metrics.auc(fpr_limited, tpr_limited) / (fpr_limited.max() - fpr_limited.min()))

def compute_mean_informedness(fpr, tpr, lowest_fpr, highest_fpr):
    fpr_limited = np.sort(fpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    tpr_limited = np.sort(tpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    return (metrics.auc(fpr_limited, tpr_limited-fpr_limited) / (fpr_limited.max() - fpr_limited.min()))

def compute_normalized_concordant_partial_area_under_curve(fpr, tpr, lowest_fpr, highest_fpr):
    fpr_limited = np.sort(fpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    tpr_limited = np.sort(tpr[(fpr > lowest_fpr) & (fpr < highest_fpr)])
    return ((metrics.auc(fpr_limited, tpr_limited) / (fpr_limited.max() - fpr_limited.min())) + (metrics.auc(tpr_limited, 1-fpr_limited) / (tpr_limited.max() - tpr_limited.min())))/2
In [10]:
import scipy.stats as stats
from scipy.interpolate import CubicSpline
from scipy.interpolate import UnivariateSpline
from statsmodels.stats.power import TTestPower
from matplotlib.patches import Patch

%config InlineBackend.figure_format = 'retina'

plt.rc('figure', dpi=500)

analysis = TTestPower()

fig = plt.figure(figsize=(30, 12),constrained_layout=True)
gs = fig.add_gridspec(3, 5)

# fig.suptitle('Comparison of classification performance', fontsize=36, y=1.06)

sample_size = 40

sample_colors = np.array(sns.color_palette("rainbow", len(sample_sizes)))

logp_threshold = -np.log10(0.05)

for ci, task in enumerate(tasks):
    for ri, method in enumerate(['spatial', 'topological', 'difference']):
        ax = fig.add_subplot(gs[ri, ci])
        
        nauc = []
        si = 2
        if sample_size == 40:
            if method ==  'difference':
                tpr = TPR[task][f'N={sample_size}']['topological'].mean(1)
                fpr = FPR[task][f'N={sample_size}']['topological'].mean(1)
                bmt = BM[task][f'N={sample_size}']['topological']
                bms = BM[task][f'N={sample_size}']['spatial']
                bmdiff = bmt - bms
                
                xlim = (min(ground_truth_effect_thresholds),max(ground_truth_effect_thresholds))

                sns.lineplot(
                    x=ground_truth_effect_thresholds,
                    y=bmdiff.mean(1),
                    style=True,
                    color=np.append(sample_colors[si]/2, 1),
                    legend=False,
                    linewidth=2,
                )

                ax.fill_between(
                    ground_truth_effect_thresholds,
                    bmdiff.mean(1) - (stats.sem(bmdiff, axis=1)*1.96),
                    bmdiff.mean(1) + (stats.sem(bmdiff, axis=1)*1.96),
                    color = np.append(sample_colors[si], 0.3),
                )

                dtof = UnivariateSpline(ground_truth_effect_thresholds, fpr, s=0)
                dtot = UnivariateSpline(ground_truth_effect_thresholds, tpr, s=0)
                dtobmd = UnivariateSpline(ground_truth_effect_thresholds, bmdiff.mean(1), s=0)
                
                nauc.append(
                    compute_normalized_partial_area_under_curve(
                        ground_truth_effect_thresholds,
                        bmdiff.mean(1),
                        get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 1),
                        get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 91282),
                    )
                )

            else:
                tpr = TPR[task][f'N={sample_size}'][method].mean(1)
                fpr = FPR[task][f'N={sample_size}'][method].mean(1)
                bm = BM[task][f'N={sample_size}'][method]

                sns.lineplot(
                    x=ground_truth_effect_thresholds,
                    y=bm.mean(1),
                    style=True,
                    color=np.append(sample_colors[si]/2, 1),
                    legend=False,
                    linewidth=2,
                )

                ax.fill_between(
                    ground_truth_effect_thresholds,
                    bm.mean(1) - (stats.sem(bm, axis=1)*1.96),
                    bm.mean(1) + (stats.sem(bm, axis=1)*1.96),
                    color = np.append(sample_colors[si], 0.3),
                )
        
                xlim = (min(ground_truth_effect_thresholds),max(ground_truth_effect_thresholds))

                dtof = UnivariateSpline(ground_truth_effect_thresholds, fpr, s=0)
                dtot = UnivariateSpline(ground_truth_effect_thresholds, tpr, s=0)
                
                nauc.append(
                    compute_normalized_partial_area_under_curve(
                        ground_truth_effect_thresholds,
                        bm.mean(1),
                        get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 1),
                        get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 91282),
                    )
                )
                
                ax.set_ylim(-0.05,1.05)
    
        leg = ax.legend(
            handles=[Patch(facecolor=np.append(sample_colors[si], 1),) for si, sample_size in [(1, 40)]],
            labels=['${:.1f}\%$'.format(100*x) for x in nauc],
            loc='upper left',
            ncol=1,
            fontsize=18,
            title="$AUC$",
            title_fontsize=24,
            labelspacing=1.,
            handlelength=1.,
        )
        leg.set_zorder(30)
        for patch in leg.get_patches():
            patch.set_width(20)
            patch.set_height(20)
            patch.set_y(-4)

        
        xlabel = ''
        if ri == 2:
            xlabel = 'Effect size threshold'
        ax.set_xlabel(xlabel, fontsize=40)

        ax.set_xlim(0.0,0.21)

        ylabel = ''
        if ci == 0:
            ylabel = '{}\nTrue Positive Rate'.format(task)
            
        if ri != 2:
            ax.set_ylim(-0.01,0.41)
        else:
            ax.set_ylim(-0.001,0.021)
        
        axylim = ax.get_ylim()
        ax.vlines(
            x=float(get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 91282)),
            ymin=axylim[0],
            ymax=axylim[1],
            linestyles='dashed',
            colors=np.array([212,24,0])/255,
            zorder = 21,
            linewidth=2,
        )
        ax.fill_between(
            [get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 91282), 0.21],
            [axylim[0], axylim[0]],
            [axylim[1], axylim[1]],
            color = [0.7,0.7,0.7,0.6],
            zorder = 20,
        )
        ax.vlines(
            x=float(get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 1)),
            ymin=axylim[0],
            ymax=axylim[1],
            linestyles='dashed',
            colors=np.array([212,24,0])/255,
            zorder = 21,
            linewidth=2,
        )
        ax.fill_between(
            [0, get_cohen_d_from_bonferroni_corrected_p_value(0.05, 1000, 1),],
            [axylim[0], axylim[0]],
            [axylim[1], axylim[1]],
            color = [0.7,0.7,0.7,0.6],
            zorder = 20,
        )
        ax.set_ylim(axylim)
        
        ax.set_facecolor(np.array([234,234,242])/255)
        ax.grid(color=(0.99,0.99,0.99,), linewidth=3)
        ax.spines['top'].set_visible(False)
        ax.spines['right'].set_visible(False)
        ax.spines['bottom'].set_visible(False)
        ax.spines['left'].set_visible(False)
        ax.tick_params(axis='both', colors=(0.5,0.5,0.5), labelcolor=(0,0,0), direction='out')

plt.show()
In [ ]: